A beaker containing liquid is placed on a table, underneath a microscope which can be moved along a vertical scale. The microscope is focused, through the liquid into a mark on the table when the reading on the scale is $a$ . It is next focused on the upper surface of the liquid and the reading is $b$ . More liquid is added and the observations are repeated. The corresponding readings are $c$ and $d$ . The refractive index of the liquid is:
A) $\dfrac{{d - b}}{{d - c - b + a}}$
B) $\dfrac{{b - d}}{{d - c - b + a}}$
C) $\dfrac{{d - c - b + a}}{{d - b}}$
D) $\dfrac{{d - b}}{{a + b - c - d}}$

Hint: You can approach the solution to the question by thinking about what the term refractive index signifies. Basically, the refractive index is the ratio of speed of light in vacuum to the speed of light in a particular medium. It can otherwise be determined as ration of real depth to apparent depth. Here, apparent depths are given in 2 cases, using which, we can reach at the value of the refractive index.

Formula Used:

Complete step by step answer:
We will be solving the question exactly as mentioned in the hint section of the solution to this question.
Let us first analyze the situation:
In the first case, we have been given two marks, $a$ and $b$ . With this, we can say that the apparent depth is $\left( {b - a} \right)$ . Using this, we can find the real height of the liquid in this case, let us call it as ${h_1}$
In the second case, liquid is added and now the two marks have become $c$ and $d$ . Making the new apparent height as $\left( {d - c} \right)$ . Using which, we can find the real height in this case as well, let us call it has ${h_2}$
We have also been told that liquid has been added, which means the real height difference is:
${h_2} - {h_1} = d - b$
Now, let us solve the question:
In first case, as we explained earlier,
${h_1} = \mu \left( {b - a} \right)$
In second case, similarly, we can say that:
${h_2} = \mu \left( {d - c} \right)$
After analyzing, we found out that:
${h_2} - {h_1} = d - b$
Putting in the values of ${h_1}$ and ${h_2}$ as we found out above, we can write:
$\mu \left( {d - c} \right) - \mu \left( {b - a} \right) = d - b$
Simplifying this, we get:
$\mu = \dfrac{{d - b}}{{d - c - b + a}}$
This value matches to that given in the option (A). Hence, option (A) is the correct answer.

Note: Many students won’t reach at the fact that the difference of the two marks given is the apparent height. Even after that, the main correlation that one has to find is that we can find the refractive index using the height difference in both the cases. This is a good trick that you should always carry around with you.